Question

A gaseous mixture of \(\mathrm{O}_{2}\) and \(\mathrm{Kr}\) has a density of \(1.104 \mathrm{~g} / \mathrm{L}\) at 435 torr and \(300 \mathrm{~K}\). What is the mole percent \(\mathrm{O}_{2}\) in the mixture?

Short answer

The mole percent of oxygen (O₂) in the gaseous mixture is approximately 68.57%.

Step-by-step solution

Gather given information and constants for the ideal gas law

We are given: - Density of mixture (d) = 1.104 g/L - Pressure (P) = 435 torr = \( \frac{435}{760} \) atm (converting to atm) - Temperature (T) = 300 K - Molar mass of O₂ = 32 g/mol - Molar mass of Kr = 83.80 g/mol - Ideal Gas Constant (R) = 0.0821 L·atm/mol·K

Calculate the total moles (n) in the mixture using the Ideal Gas Law

The Ideal Gas Law equation is given by: PV = nRT Where: - P: Pressure - V: Volume - n: Moles - R: Ideal Gas Constant - T: Temperature We want to find the total moles (n) for the gaseous mixture: n = \(\frac{PV}{RT}\)

Express the density in terms of molar mass and mole fractions

The density of the gas mixture can be expressed in terms of the total moles and molar mass: d = \(\frac{n(M_{O_{2}}x_{O_{2}} + M_{Kr}x_{Kr})}{V}\) 1.104 g/L = \(\frac{(M_{O_{2}}x_{O_{2}} + M_{Kr}x_{Kr})(PV)}{R(T)}\) Where: - M: molar mass - x: mole fraction We can now solve for the mole fraction x: \(x_{O_{2}}=\frac{(1.104)(0.0821)(300)-(PV)(M_{Kr})}{(PV)(M_{O_{2}}-M_{Kr})}\)

Calculate the mole percent of O₂ in the mixture

Finally, we can plug in the given values and calculate the mole fraction (x) of O₂ and then convert that to the mole percent of O₂: \(x_{O_{2}}=\frac{(1.104)(0.0821)(300)-( \frac{435}{760})(M_{Kr})}{(\frac{435}{760})(M_{O_{2}}-M_{Kr})}\) \(x_{O_{2}}=\frac{(1.104)(0.0821)(300)-( \frac{435}{760})(83.80)}{(\frac{435}{760})(32-83.80)}\) \(x_{O_{2}} \approx 0.6857\) Mole percent of O₂ = \(0.6857 × 100\% = 68.57 \% \) The mole percent of oxygen (O₂) in the gaseous mixture is approximately 68.57%.

Key concepts

Understanding the Ideal Gas Law

In the realm of chemistry, the ideal gas law is a fundamental equation that relates the pressure (P), volume (V), number of moles of gas (n), and temperature (T) of an ideal gas to each other. Expressed as PV = nRT, this law is instrumental in determining properties of gases under various conditions. The ideal gas constant (R) is a universal value used to bridge these properties, and it's crucial to use the correct units for R depending on the other units in the equation.

The ideal gas law assumes that gas particles are point particles that experience no intermolecular forces and occupy no volume. Although this is not true for real gases, the ideal gas law provides a close approximation for the behavior of real gases under many conditions. When dealing with gas mixtures, as in our problem, the ideal gas law can still apply; the total pressure exerted by the mixture is the result of the sum of partial pressures of each component gas. Understanding how to manipulate and apply the ideal gas law is key to solving many problems in chemistry, including calculating molar mass, determining gas density, or finding mole fractions in a gas mixture.

The Significance of Molar Mass

Molar mass, symbolized as M and measured in grams per mole (g/mol), represents the mass of one mole of a substance. It serves as a bridge connecting the microscopic world of atoms and molecules with the macroscopic world we can measure. The molar mass of a gas is particularly important when we use the ideal gas law. By knowing the molar mass of a gas or each component in a gas mixture, we can determine the mass of the gas present in a known volume, facilitating the calculation of gas densities and mole fractions.

Moreover, molar masses of component gases in a mixture are foundational when determining the mole fraction and, consequently, the mole percent of each component. In our calculation, the molar masses for oxygen (O₂) and krypton (Kr) are used to relate the mass of the gas mixture with the total number of moles. These values are critical for the accurate calculation of mole fractions, which lead to the calculation of mole percents in gas mixtures.

Calculating Mole Fraction

Mole fraction, denoted as x, is a dimensionless quantity that represents the ratio of the number of moles of a component to the total number of moles in a mixture. It is a way to express the composition of a mixture without the need for masses or volumes, which can be especially useful when considering the behavior of gases. To find the mole fraction of a gas in a mixture, one must know or be able to calculate the total number of moles in the mixture, as well as the moles of the component of interest.

In gas mixtures, such as the O₂ and Kr example, the mole fraction is essential in deducing the percentage composition by volume or by the number of particles. As the mole fraction is a measure of proportionality, it allows us to determine how much of one gas is present compared to another within a mixture. Calculating the mole fraction is a key step that precedes the final calculation of the mole percent, which conveys the percentage of a particular gas component in the mix.

Determining Gas Mixture Density

Density, represented by d and measured in units of mass per volume (such as g/L), is an intrinsic property that provides insight into the composition and concentration of a substance. For gas mixtures, density can be determined by the masses and mole fractions of the component gases as well as the conditions of pressure and temperature the gas experiences. It’s a composite property that can be calculated once the molar masses and the proportions of gases in a mixture are known.

In the context of our problem, gas mixture density helps us unlock the quantities needed to determine the mole percent of oxygen. The provided density of the gaseous mixture, along with the conditions described by the ideal gas law, allows us to set up an equation wherein the only unknowns are the mole fractions of the gases involved. By solving for these values, we establish a direct link to the concept of mole percent. Density thus acts as an essential variable that can reveal much about the composition of a gas mixture when the proper equations and constants are applied effectively.